Abstract

The mutual coherence function that appears in the calculation of wave propagation in a turbulent atmosphere has never to our knowledge been derived in the Rytov approximation without appeal to the principle of energy conservation. We show why the Rytov approximation is not applicable beyond the first-order terms. A systematic perturbation based on the exponentiation of the Born series is carried out. Not only is energy conservation automatically satisfied in the new series, but also the results are consistent with other approaches.

© 1981 Optical Society of America

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References

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  1. V. I. Tartarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  2. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57 (1967).
    [CrossRef]
  3. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399 (1972).
    [CrossRef] [PubMed]
  4. A. Ishimara, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295 (1969).
    [CrossRef]
  5. We follow the argument in Ref. 2 that the crossing correlation of the amplitude and the phase can be neglected.
  6. This relationship is also essential for derivation of the intensity–intensity correlation function. See Z. I. Felzlin, Yu A. Kravison, “Broadening of a laser beam in a turbulent medium,” Sov. Radiophys. 10, 68 (1967). They referred Eq. (2) to Tartarski's doctoral dissertation.
  7. W. P. Brown, “Second moment of a wave propagating in a random medium,” J. Opt. Soc. Am. 61, 1051 (1971).
    [CrossRef]
  8. M. Beran, “Propagation of a finite beam in a random medium,” J. Opt. Soc. Am. 60, 518 (1970).
    [CrossRef]
  9. A. Ishimara, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  10. H. T. Yura, “Optical propagation through a turbulent medium,” J. Opt. Soc. Am. 59, 111 (1969).
    [CrossRef]
  11. R. A. Schmeltzer, “Laser beam propagation,” Q. Appl. Math. 24, 339 (1966).

1972 (1)

1971 (1)

1970 (1)

1969 (2)

H. T. Yura, “Optical propagation through a turbulent medium,” J. Opt. Soc. Am. 59, 111 (1969).
[CrossRef]

A. Ishimara, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295 (1969).
[CrossRef]

1967 (2)

This relationship is also essential for derivation of the intensity–intensity correlation function. See Z. I. Felzlin, Yu A. Kravison, “Broadening of a laser beam in a turbulent medium,” Sov. Radiophys. 10, 68 (1967). They referred Eq. (2) to Tartarski's doctoral dissertation.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57 (1967).
[CrossRef]

1966 (1)

R. A. Schmeltzer, “Laser beam propagation,” Q. Appl. Math. 24, 339 (1966).

Beran, M.

Brown, W. P.

Felzlin, Z. I.

This relationship is also essential for derivation of the intensity–intensity correlation function. See Z. I. Felzlin, Yu A. Kravison, “Broadening of a laser beam in a turbulent medium,” Sov. Radiophys. 10, 68 (1967). They referred Eq. (2) to Tartarski's doctoral dissertation.

Fried, D. L.

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57 (1967).
[CrossRef]

Ishimara, A.

A. Ishimara, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295 (1969).
[CrossRef]

A. Ishimara, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Kravison, Yu A.

This relationship is also essential for derivation of the intensity–intensity correlation function. See Z. I. Felzlin, Yu A. Kravison, “Broadening of a laser beam in a turbulent medium,” Sov. Radiophys. 10, 68 (1967). They referred Eq. (2) to Tartarski's doctoral dissertation.

Schmeltzer, R. A.

R. A. Schmeltzer, “Laser beam propagation,” Q. Appl. Math. 24, 339 (1966).

Tartarski, V. I.

V. I. Tartarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Yura, H. T.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57 (1967).
[CrossRef]

Q. Appl. Math. (1)

R. A. Schmeltzer, “Laser beam propagation,” Q. Appl. Math. 24, 339 (1966).

Radio Sci. (1)

A. Ishimara, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295 (1969).
[CrossRef]

Sov. Radiophys. (1)

This relationship is also essential for derivation of the intensity–intensity correlation function. See Z. I. Felzlin, Yu A. Kravison, “Broadening of a laser beam in a turbulent medium,” Sov. Radiophys. 10, 68 (1967). They referred Eq. (2) to Tartarski's doctoral dissertation.

Other (3)

We follow the argument in Ref. 2 that the crossing correlation of the amplitude and the phase can be neglected.

V. I. Tartarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

A. Ishimara, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

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Equations (20)

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MCF = G ( x 0 , x ) G ( x 0 , x ) = G 0 ( x 0 , x ) G 0 ( x 0 , x ) exp [ 1 / 2 ( D s + D A ) ] ,
χ 2 = χ 1 2
[ 2 + k 2 ( 1 + 2 n 1 ) ] G ( x 0 , x ) = 4 π δ ( x 0 x ) ,
G = exp ( ϕ 0 + ϕ 1 + ϕ 2 ) ,
G 0 e ϕ 0 exp [ i k | x 0 x | ] | x 0 x | ,
2 ϕ 1 + 2 ϕ 0 ϕ 1 + 2 k 2 n 1 = 0 ,
2 ϕ 2 + 2 ϕ 0 ϕ 2 + ϕ 1 ϕ 1 = 0 .
| x 0 x | exp ( i k | x 0 x | ) ( 2 W 1 + k 2 W 1 ) = 2 k 2 n 1 ( x ) .
W 1 = 2 k 2 G 0 ( x 0 , x ) n 1 ( x ) G 0 ( x , x ) d 3 x / 4 π ,
ϕ 1 ϕ 1 | x = x 0 = ( 2 k 2 4 π ) 2 lim x x 0 G 0 ( x 0 , x ) n 1 ( x ) × G 0 ( x , x ) d 3 x 0 .
| x x 0 | exp ( i k | x x 0 | ) ( 2 W 2 + k 2 W 2 ) = ϕ 1 ϕ 1 , |
G = G 0 + U 1 + U 2 ;
( 2 + k 2 ) U 1 + 2 k 2 n 1 ( x ) G 0 ( x 0 , x ) = 0 ,
( 2 + k 2 ) U 2 + 2 k 2 n 1 ( x ) U 1 = 0 .
U 1 = 2 k 2 4 π G 0 ( x 0 , x ) n 1 ( x ) G 0 ( x , x ) d 3 x ,
U 2 = ( 2 k 2 4 π ) 2 G 0 ( x 0 , x ) n 1 ( x ) d 3 x × G 0 ( x , x ) n 1 ( x ) G 0 ( x , x ) d 3 x .
G = G 0 exp ( f 1 + f 2 + f 3 ) ,
f 1 = U 1 / G 0 , f 2 = U 2 / G 0 U 1 2 / 2 G 0 2 , f 3 = U 3 / G 0 U 1 U 2 / G 0 2 U 1 3 / 6 G 0 3 .
MCF = exp [ ( ϕ 0 + ϕ 0 * ) + ( f 1 + f 1 * ) 2 / 2 + f 2 + f 2 * ] ,
U 2 / G 0 = 1 / 2 | U 1 / G 0 | 2 = 4 π 2 k 2 z Φ ( p ) p d p z α ,

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